Non-Asymptotic Bounds via the Nussbaum-Szkoła Mapping for Quantum Hypothesis Testing

Abstract

Quantum hypothesis testing concerns the discrimination between quantum states. This paper introduces novel lower and upper bounds for asymmetric quantum hypothesis testing based on the Nussbaum-Szkoła mapping. Together, these bounds provide a unified recovery of both converse and achievability results across all major asymptotic regimes, including large-, moderate-, and small-deviations. Unlike existing bounds, which either rely on technically involved information-spectrum arguments or suffer from fixed prefactors and limited applicability in the non-asymptotic regime, the proposed bounds arise from a unified framework and enable, in some cases, the direct use of classical results. It is further demonstrated that these bounds provide accurate approximations to the optimal quantum error trade-off function at small blocklengths. Numerical comparisons with existing bounds, including those based on fidelity and information spectrum methods, highlight their improved tightness.

The paper studies non-asymptotic bounds on the optimal type-II error $\beta_\alpha(\rho,\sigma)$ under a type-I error constraint $\alpha$, using the classical Nussbaum–Szkoła distributions $P,Q$ associated with the states’ spectral decompositions. Its main general converse is $\beta_\alpha(\rho,\sigma)\ge s\,\beta_{\alpha/(1-s)}(P,Q)$ for $0\le s<1$ and $\alpha\le1-s$. The proof combines the variational characterization of the quantum testing trade-off with a weighted Cauchy–Schwarz inequality. Together with the existing achievability bound $\beta_\alpha(\rho,\sigma)\le s^{-1}\beta_{(1-s)\alpha}(P,Q)$, this gives a classical–quantum comparison with adjustable error rescalings and prefactors. The converse is shown to be asymptotically tight for nearly orthogonal pure states. When $\rho$ is pure, a stronger achievability bound is established: $\beta_\alpha(\rho,\sigma)\le s_1^{-1}\beta_{s_2\alpha}(P,Q)$, with $s_1\le1$ for $s_2\le1/2$ and $s_1\le4s_2(1-s_2)$ otherwise. A rank-one perturbation analysis yields this region and a sharp universal factor $4/3$ comparing the quantum weighted minimum with its Nussbaum–Szkoła counterpart in the pure–mixed setting. Choosing blocklength-dependent parameters recovers known second-order asymptotics with $O(\log n)$ remainder, moderate-deviation expansions, and Hoeffding exponents from classical results. Numerical comparisons use five-copy qubit examples. The paper notes that existing harmonic-mean bounds are tighter in the mixed–mixed examples and recover the exact third-order term, whereas the proposed bounds emphasize analytic tractability. The improved achievability bound requires a pure null state; the asymptotic analysis assumes $\operatorname{supp}(\rho)\subseteq\operatorname{supp}(\sigma)$.